duality in convex geometry…
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DUALITY IN CONVEX GEOMETRY…. WELL, GEOMETRIC TOMOGRAPHY, ACTUALLY!. Richard Gardner. Scope of Geometric Tomography. Computerized tomography. Discrete tomography. Robot vision. Convex geometry. Point X-rays. Parallel X-rays. Imaging. Projections; classical Brunn-Minkowski theory. - PowerPoint PPT PresentationTRANSCRIPT
04/22/23 1
DUALITY IN CONVEX GEOMETRY…DUALITY IN CONVEX GEOMETRY…
Richard Gardner
WELL, GEOMETRIC TOMOGRAPHY, ACTUALLY!WELL, GEOMETRIC TOMOGRAPHY, ACTUALLY!
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Scope of Geometric TomographyScope of Geometric Tomography
Point X-rays
Integral geometry
Local theory of Banach spaces
Robot vision
Stereology and local stereology
Computerized tomographyDiscrete tomography
Parallel X-rays
Minkowski geometry
Imaging
Pattern recognition
Convex geometry
Sections through a fixed point; dual Brunn-Minkowski theory
Projections; classical Brunn-Minkowski theory
?
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ProjectionsProjections and and SectionsSections Theorem 1. (Blaschke and Hessenberg, 1917.) Suppose that 2 ≤ k ≤ n -1, and every k-projection K|S of a compact convexset in Rn is a k-dimensional ellipsoid. Then K is an ellipsoid. Theorem 1'. (Busemann, 1955.) Suppose that 2 ≤ k ≤ n -1, and every k-section K ∩ S of a compact convex set in Rn containing the origin in its relative interior is a k-dimensional ellipsoid. Then K is an ellipsoid. Busemann’s proof is direct, but Theorem 1' can be obtained fromTheorem 1 by using the (not quite trivial) fact that the polar of
anellipsoid containing the origin in its interior is an ellipsoid.Burton, 1976: Theorem 1' holds for arbitrary compact convex setsand hence for arbitrary sets.
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The The SupportSupport FunctionFunction
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The The RadialRadial FunctionFunction and and Star BodiesStar Bodies
E. Lutwak, Dual mixed volumes, Pacific J. Math. 58 (1975),
531-538.
Star body: Body star-shaped at o whose radial function is positive and continuous,OR one of several other alternative definitions in the literature!
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Polarity – a Projective NotionPolarity – a Projective Notion
Let K be a convex body in Rn with o in int K, and let φ be a nonsingular projective transformation of Rn, permissible for K, such that o is in int φK. Then there is a nonsingular projective transformation ψ, permissible for K*, such that (φK)* = ψK*.
P. McMullen and G. C. Shephard, Convex polytopes and the upper bound conjecture, Cambridge University Press, Cambridge, 1971.
).(/1)(* uhu KK
For a convex body K containing the origin in its interior and u in Sn-1,
Then, if S is a subspace,
we have .*)|( * SKSK
We have (φK)* = φ–tK*, for φ in GLn, but…
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Polarity Usually FailsPolarity Usually Fails Theorem 2. (Süss, Nakajima, 1932.) Suppose that 2 ≤ k ≤ n -1 andthat K and L are compact convex sets in Rn. If all k-projections K|Sand L|S of K and L are homothetic (or translates), then K and L arehomothetic (or translates, respectively). Theorem 2'. (Rogers, 1965.) Suppose that 2 ≤ k ≤ n -1 andthat K and L are compact convex sets in Rn containing the origin intheir relative interiors. If all k-sections K ∩ S and L ∩ S of K and Lare homothetic (or translates), then K and L are homothetic (ortranslates, respectively).
Problem 1. (G, Problem 7.1.) Does Theorem 2' hold for star bodies,or perhaps more generally still?
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WidthWidth and and BrightnessBrightness
width function
)()(
)()(
uhuh
lKVuw
KK
uK
)()( uKVubK
brightness function
uK
u
u
K
K
KSu
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Aleksandrov’s Projection TheoremAleksandrov’s Projection Theorem
For o-symmetric convex bodies K and L,
1
),(|21
nS
vKdSv|u(u)bK
. LKu(u)b(u)b LK
Cauchy’s projection formula
Cosine transform of surface area measure of K
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Funk-Minkowski Section TheoremFunk-Minkowski Section Theorem
)()( uKVusK
. LKu(u)s(u)s LK
Funk-Minkowski section theorem: For o-symmetric star bodies K and L,
section function u
K
uS
nK
n
dvvn 1
1)(1
1
Spherical Radon transform of (n-1)st power of radial function of K
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Projection BodiesProjection Bodies
KK bh
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Shephard’s ProblemShephard’s Problem
• Petty, Schneider, 1967: For o-symmetric convex bodies K and L,
(i) if L is a projection body and (ii) if and only if n = 2. )()( LVKVu(u)b(u)b LK
K L
A counterexample for n = 3
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Intersection BodiesIntersection Bodies
KIK sErwin Lutwak
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Busemann-Petty Problem Busemann-Petty Problem (Star Body Version)(Star Body Version)
• Lutwak,1988: For o-symmetric star bodies K and L,
(i) if K is an intersection body and (ii) if and only if n = 2. )()( LVKVu(u)s(u)s LK
Hadwiger,1968: A counterexample for n = 3
K L
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Minkowski Minkowski and and RadialRadial Addition Addition
LK ~LK
LL
K K
22 ),(),(2),()( tLLVstLKVsKKVtLsKV 2
~~2
~),(),(2),()~( tLLVstLKVsKKVtLsKV
LKLK hhh LKLK ~
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Dual Mixed VolumesDual Mixed Volumes
1
21.)()()(1),...,,( 21
~
nn
SKKKn duuuu
nKKKV
The dual mixed volume of star bodies K1, K2,…, Kn is
E. Lutwak, Dual mixed volumes, Pacific J. Math. 58 (1975), 531-
538.
.)(),(
),;,(),(
1)(),(
inGi
ni dSSKV
incinBiKV
incKV
.)()(1 ),;,()(),(
~~
1
inGi
n
S
iKi dSSKVduu
ninBiKVKV
n
Intrinsic volumes and Kubota’s formula:
Dual volumes and the dual Kubota formula (Lutwak, 1979):
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Characterizations of Characterizations of Projection Projection andand Intersection Intersection
Bodies Bodies II Firey, 1965, Lindquist, 1968: A convex body K in Rn is a projection body iff it is a limit (in
the Hausdorff metric) of finite Minkowski sums of n-dimensional o-symmetric ellipsoids.
Definition: A star body is an intersection body if ρL = Rμ for some
finite even Borel measure μ in Sn-1.
Goodey and Weil, 1995: A star body K in Rn is an intersection body iff it is a limit (in the radial metric) of finite radial sums of n-dimensional o-symmetric ellipsoids.
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Characterizations of Characterizations of Projection Projection andand Intersection Intersection
Bodies Bodies IIII Weil, 1976: Let L be an o-symmetric convex body in Rn. An o-symmetric convex body K in Rn is a projection body iff
for all o-symmetric convex bodies M such that ΠL is contained in ΠM.),;1,();1,( KnMVKnLV
For a unified treatment, see:
Zhang, 1994: Let L be an o-symmetric star body in Rn. An o-symmetric star body K in Rn is an intersection body iff
),;1,();1,(~~
KnMVKnLV for all o-symmetric star bodies M such that IL is contained in IM.
P. Goodey, E. Lutwak, and W. Weil, Functional analytic characterizations ofclasses of convex bodies, Math. Z. 222 (1996), 363-381.
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Lutwak’s DictionaryLutwak’s DictionaryConvex bodies Star bodiesProjections Sections through o
Support function hK Radial function ρK
Brightness function bK Section function sK
Projection body ΠK Intersection body IK
Cosine transform Spherical Radon transform
Surface area measure ρKn-1
Mixed volumes Dual mixed volumes
Brunn-Minkowski ineq. Dual B-M inequalityAleksandrov-Fenchel Dual A-F inequality
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Is Lutwak’s Dictionary Infallible?Is Lutwak’s Dictionary Infallible? Theorem 3. (Goodey, Schneider, and Weil,1997.) Most convex
bodies in Rn are determined, among all convex bodies, up to translation and reflection in o, by their width and brightness functions.
Theorem 3'. (R.J.G., Soranzo, and Volčič, 1999.) The set of star bodies in Rn that are determined, among all star bodies, up to reflection in o, by their k-section functions for all k, is nowhere dense.
Notice that this (very rare) phenomenon does not apply within the class of o-symmetric bodies. There I have no example of this sort.
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Two Two Unnatural Unnatural ProblemsProblems 1. (Busemann, Petty, 1956.) If K and L are o-symmetric convex
bodies in Rn such that sK(u) ≤ sL(u) for all u in Sn-1, is V(K) ≤ V(L)?
2. Problem 2. (R.J.G., 1995; G, Problem 7.6.) If K and L are o-symmetric star bodies in R3 whose sections by every plane through o have equal perimeters, is K =L?
Solved: Yes, n = 2 (trivial), n = 3 (R.J.G., 1994), n = 4 (Zhang,1999).
No, n ≥ 5 (Papadimitrakis, 1992, R.J.G., Zhang, 1994). Unified: R.J.G., Koldobsky, and Schlumprecht, 1999.
Unsolved.
Theorem 4. (Aleksandrov, 1937.) If K and L are o-symmetric convex bodies in R3 whose projections onto every plane have equal perimeters, then K =L.
SS
K duuSBSKV2
)(21),(
~
Yes. R.J.G. and Volčič, 1994.
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Aleksandrov-FenchelAleksandrov-Fenchel InequalityInequality If K1, K2,…, Kn are compact convex sets in Rn, then
.),...,;,(),...,,(1
121
i
jnij
in KKiKVKKKV
Dual Aleksandrov-Fenchel InequalityDual Aleksandrov-Fenchel Inequality If K1, K2,…, Kn are star bodies in Rn, then
,),...,;,(),...,,(1
1
~
21
~
i
jnij
in KKiKVKKKV
with equality when if and only if K1, K2,…, Ki are dilatates of each other.
ni
Proof of dual Aleksandrov-Fenchel inequality follows directly from an extension of Hölder’s inequality.
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Brunn-MinkowskiBrunn-Minkowski inequalityinequality If K and L are convex bodies in Rn, then
with equality iff K and L are homothetic.
nnn LVKVLKV /1/1/1 )()()( nnn LVKVLnKV /1/)1( )()();1,(
and (B-M)
(M1)
Dual Brunn-Minkowski Dual Brunn-Minkowski inequalityinequality If K and L are star bodies in Rn, then
nnn LVKVLKV /1/1/1 )()()~( and
nnn LVKVLnKV /1/)1(~
)()();1,(
with equality iff K and L are dilatates.
(d.B-M)
(d.M1)
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RelationsRelations
.02)~(1);1,( /1~
nLKVLnKV
.1)()( LVKV Suppose that
(B-M)→(M1)
(d.B-M)→(d.M1)
.02)(1);1,( /1 nLKVLnKV H. Groemer, On an inequality of Minkowski for mixed
volumes, Geom. Dedicata. 33 (1990), 117-122.
R.J.G. and S. Vassallo, J. Math. Anal. Appl. 231 (1999), 568-587 and 245 (2000), 502-512.
.1);1,(2)~()12(
1 ~
1
LnKVLKV
nn n
n
.1);1,(2)(1
LnKVLKV
nn
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Busemann Intersection InequalityBusemann Intersection Inequality
with equality if and only if K is an o-symmetric ellipsoid.
If K is a convex body in Rn containing the origin in its interior, then
,)()( 121
n
nn
nn KVIKV
H. Busemann, Volume in terms of concurrent cross-sections,
Pacific J. Math. 3 (1953), 1-12.
1
)(1)(nS
n duuKVn
IKV
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Generalized BusemannGeneralized BusemannIntersection InequalityIntersection Inequality
with equality when 1 < i < n if and only if K is an o-symmetric ellipsoid and when i = 1 if and only if K is an o-symmetric star body, modulo sets of measure zero.
If K is a bounded Borel set in Rn and 1 ≤ i ≤ n, then
,)()(),(
ini
n
ni
inG
ni KVdSSKV
H. Busemann and E. Straus, 1960; E. Grinberg, 1991; R. E. Pfiefer,
1990; R.J.G., Vedel Jensen, and Volčič, 2003. R.J.G., The dual Brunn-Minkowski theory for bounded Borel sets: Dual
affine quermassintegrals and inequalities, Adv. Math. 216 (2007), 358-386.
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Petty’s ConjecturedPetty’s ConjecturedProjection InequalityProjection Inequality
with equality if and only if K is an ellipsoid?
Let K be a convex body in Rn. Is it true that
,)()( 121
n
nn
nn KVKV
C. Petty, 1971.
.)()( 1
1
* nn
n
n KVKV
Petty Projection Inequality:
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Other RemarksOther Remarks• “Bridges”: Polar projection bodies, centroid bodies, p-
cosine transform, Fourier transform,…
• The Lp-Brunn-Minkowski theory and beyond…
• Gauss measure, p-capacity, etc.
• Valuation theory…
• There may be one all-encompassing duality, but perhaps more likely, several overlapping dualities, each providing part of the picture.